Introduction to Topology and Geometry

Saul Stahl

Saul Stahl's new Introduction to Topology and Geometry is not for the casual reader. In about 400 pages, liberally illustrated, Stahl provides (not in this order) a crash course in differential geometry, a look at hyperbolic geometry, a primer on the basics of topology — including the fundamental group, as well as a discussion of graphs and surfaces and knots and links. This is quite a bit to chew on, considering that the upshot of the chapter on differential geometry is Gauss's Total Curvature Theorem, that the discussion of the fundamental group not only includes many computations, but leads to the knot group and a short note on the Poincaré conjecture.

The preface to the book begins, "This book is intended to serve as a text for a two-semester undergraduate course in topology and modern geometry. It is devoted almost entirely to the geometry of the last two centuries.... Much of the material here has traditionally been part of the realm of graduate mathematics, and its persentation in undergraduate courses necessitates the adoption of certain informalities that would be unacceptable at the more advanced levels."

There is a lot here to like. This is a nice collection of topics that are not usually present in a single text. And there are many examples and pictures included in the exposition. But as the author states in his preface regarding the chapters on differential geometry "The expostion is as elementary as the author could make it and still meet his goals: explanations of Gauss's Total Curvature Theorem and hyperbolic geometry." I think readers may feel rushed, here and elsewhere in the text. If I were teaching a class from it, I think I would be unpacking the book quite a bit. I can also only envision myself having a single semester in which to teach a course focusing on topics such as these, and then I think I would feel conflicted about using this book; there are better texts for treating just topology, just geometry, or just graph theory.

That said, I am glad that the book is on my bookshelf. Stahl has worked hard to make these ideas accessible to undergraduates. I will be looking to him for ideas when I want to speak on some of these topics in a class; and for that bright, motivated student looking for something extra to do, it will provide a great resource. And if I did have two semesters rather than one in which to teach such a course, I might be persuaded to give the book a spin.